Finite Group Databasebeta

← #238138

Subgroups — #238138

Order 512.

1068 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34938C211yesmaximal, characteristic
2#21804C211yesmaximal, characteristic
3#21313C211yesmaximal, characteristic
4C2×#1987C211yesmaximal, characteristic
5#10355C214yesmaximal, complemented
6#10349C214yesmaximal, complemented
7#8045C211yesmaximal, characteristic
8#7901C211yesmaximal, characteristic
9#7891C211yesmaximal, characteristic
10C2×#533—21no—
11C22×C4⋊Q8C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#423C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15#2552C2212yes—
16#2551C2212yes—
17#2550—22no—
18#2549—22no—
19#2440—22no—
20#2399—22no—
21C4×C4⋊Q8—22no—
22#2296—21no—
23#2285C2211yescharacteristic
24C4×#117C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26#1988—22no—
27#1987—22no—
28#1987C2212yes—
29#1987C2212yes—
30#1987C2214yes—
31#1981—22no—
32#1956C2214yes—
33#1787—22no—
34#1736—21no—
35#1736—21no—
36C2×#340—22no—
37#1296C2214yes—
38#1279C2212yes—
39#1279C2214yes—
40#1279C2212yes—
41C23×Q8—21no—
42C22×M16—21no—
43C23×C8—21no—
44#533—44no—
45#533—44no—
46C2×C4⋊Q8—22no—
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8Q814yescomplemented
55C2×C4⋊Q8C2312yes—
56C2×#125—22no—
57C2×C22⋊Q8—24no—
58C2×C4×Q8—24no—
59C2×#117—21no—
60C2×#117D411yescharacteristic
61C2×#117D411yescharacteristic
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4D412yes—
65C22×C42—21no—
66C22×C42C2311yescharacteristic
67#431—22no—
68#431—22no—
69#423—22no—
70#423—22no—
71C2×C4⋊C8—22no—
72C2×C4⋊C8—22no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8C2312yes—
77C2×C4⋊C8C2312yes—
78C2×#103—28no—
79C2×C22⋊C8—21no—
80C2×C4×C8C2312yes—
81#398—24no—
82#398—24no—
83#395—24no—
84#395—24no—
85#391—24no—
86#389—24no—
87#389—22no—
88#389—22no—
89#389—22no—
90#389—22no—
91#385—24no—
92#384—22no—
93#384C2312yes—
94#382—21no—
95#380—21no—
96#380D412yes—
97C4×C4⋊C4—24no—
98C4×C4⋊C4—21no—
99C4×C4⋊C4—21no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4D412yes—
102C4×C4⋊C4D412yes—
103C4×C4⋊C4D412yes—
104C4×C4⋊C4C2311yescharacteristic
105C4×C4⋊C4C2311yescharacteristic
106C4×C4⋊C4C2312yes—
107C4×C22⋊C4—21no—
108C4×C22⋊C4—21no—
109#376—21no—
110C2×#95—21no—
111C43—21no—
112#340—24no—
113#340—24no—
114#340—24no—
115#340—24no—
116C22×Q8—24no—
117C22×Q8—24no—
118C22×Q8—22no—
119C22×Q8—41no—
120C22×Q8—42no—
121C22×Q8—84no—
122C23×C4—22no—
123C23×C4—41no—
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C2×M16—42no—
128C2×M16—42no—
129C2×M16—42no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—44no—
134C22×C8—42no—
135C4⋊Q8—24no—
136C4⋊Q8—24no—
137C4⋊Q8—24no—
138C4⋊Q8—22no—
139C4⋊Q8—22no—
140C4⋊Q8—22no—
141C4⋊Q8—44no—
142C4⋊Q8—44no—
143C4⋊Q8—44no—
144C4⋊Q8—44no—
145C4⋊Q8—42no—
146C4⋊Q8—42no—
147C4⋊Q8—42no—
148C4⋊Q8C2×Q814yes—
149#125—44no—
150#125—44no—
151C22⋊Q8—416no—
152C4×Q8—416no—
153#117—22no—
154#117—22no—
155#117—42no—
156#117—42no—
157#117SD1614yes—
158#117Q1614yes—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4C2×D412yes—
178C2×C4⋊C4C2×D412yes—
179C2×C4⋊C4C2×D412yes—
180C2×C4⋊C4C2×D412yes—
181C2×C4⋊C4C2×D414yes—
182C2×C22⋊C4—21no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—22no—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—21no—
190C2×C42—21no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—42no—
194C2×C42—42no—
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42C2×Q812yes—
199C2×C42#4112yes—
200C2×C42#4112yes—
201C2×C42#4112yes—
202C2×C42C2411yescharacteristic
203C4⋊C8—22no—
204C4⋊C8—22no—
205C4⋊C8—22no—
206C4⋊C8—22no—
207C4⋊C8—42no—
208C4⋊C8—42no—
209C4⋊C8—42no—
210#103—88no—
211C22⋊C8—42no—
212C4×C8—22no—
213C4×C8—22no—
214#95—24no—
215#95—24no—
216#95—22no—
217#95—22no—
218#95—22no—
219#95—22no—
220#95—21no—
221#95—21no—
222#95—22no—
223C24#12011yescharacteristic
224C2×Q8—24no—
225C2×Q8—48no—
226C2×Q8—48no—
227C2×Q8—48no—
228C2×Q8—42no—
229C2×Q8—44no—
230C2×Q8—84no—
231C2×Q8—84no—
232C2×Q8—88no—
233C2×Q8—88no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—24no—
241C22×C4—42no—
242C22×C4—44no—
243C22×C4—42no—
244C22×C4—44no—
245C22×C4—44no—
246C22×C4—82no—
247C22×C4—84no—
248C22×C4#12011yescharacteristic
249C22×C4#12011yescharacteristic
250C22×C4#12011yescharacteristic
251C22×C4C22⋊Q812yes—
252C22×C4C22⋊Q812yes—
253C22×C4C22⋊Q812yes—
254C22×C4#12412yes—
255C22×C4#12412yes—
256C22×C4#12412yes—
257C22×C4C2×SD1612yes—
258C22×C4C2×Q1612yes—
259C22×C4AGL(1,ℤ/8)12yes—
260C22×C4#13712yes—
261C22×C4C22×D411yescharacteristic
262C22×C4C22×D411yescharacteristic
263C22×C4C22×D411yescharacteristic
264M16—84no—
265C2×C8—44no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—44no—
272C2×C8—84no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—24no—
277C4⋊C4—24no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—44no—
281C4⋊C4—44no—
282C4⋊C4—48no—
283C4⋊C4—48no—
284C4⋊C4—48no—
285C4⋊C4—48no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—44no—
293C4⋊C4—44no—
294C4⋊C4—84no—
295C4⋊C4C2×SD1614yes—
296C4⋊C4C2×Q1614yes—
297C4⋊C4#13714yes—
298C22⋊C4—24no—
299C22⋊C4—24no—
300C22⋊C4—44no—
301C42—24no—
302C42—24no—
303C42—41no—
304C42—41no—
305C42—41no—
306C42—41no—
307C42—42no—
308C42—42no—
309C42—44no—
310C42—44no—
311C42—82no—
312C42C22⋊Q812yes—
313C42C22⋊Q812yes—
314C42C22⋊Q812yes—
315C42#12412yes—
316C42#12412yes—
317C42#12412yes—
318C42C22×Q811yescharacteristic
319C42C2×#4111yescharacteristic
320C42C2×#4111yescharacteristic
321C42C2×#4111yescharacteristic
322C23—22no—
323C23—22no—
324C23#39312yes—
325C23#45012yes—
326C23#45112yes—
327C23C2×#12011yescharacteristic
328Q8—48no—
329Q8—84no—
330Q8—88no—
331Q8—816no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—24no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—44no—
350C2×C4—42no—
351C2×C4—44no—
352C2×C4—42no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—42no—
356C2×C4—88no—
357C2×C4—88no—
358C2×C4—84no—
359C2×C4#39312yes—
360C2×C4#39312yes—
361C2×C4#39312yes—
362C2×C4#45012yes—
363C2×C4#45112yes—
364C2×C4#47514yes—
365C2×C4#47714yes—
366C2×C4#48714yes—
367C2×C4#48914yes—
368C2×C4C2×C22⋊Q811yescharacteristic
369C2×C4C2×C22⋊Q811yescharacteristic
370C2×C4C2×C22⋊Q811yescharacteristic
371C2×C4C2×#12411yescharacteristic
372C2×C4C2×#12411yescharacteristic
373C2×C4C2×#12411yescharacteristic
374C2×C4C22×SD1611yescharacteristic
375C2×C4C22×Q1611yescharacteristic
376C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
377C2×C4C2×#13711yescharacteristic
378C2×C4C2×#13712yes—
379C2×C4#57512yes—
380C8—82no—
381C8—82no—
382C8—82no—
383C22—24no—
384C22—24no—
385C22—22no—
386C22—22no—
387C22#202414yes—
388C22C2×#39311yescharacteristic
389C22C2×#45011yescharacteristic
390C22C2×#45111yescharacteristic
391C22#300712yes—
392C22#302112yes—
393C4—22no—
394C4—21no—
395C4—21no—
396C4—22no—
397C4—44no—
398C4—44no—
399C4—44no—
400C4—82no—
401C4—84no—
402C4—84no—
403C4#202414yes—
404C4C2×#47511yescharacteristic
405C4C2×#47711yescharacteristic
406C4#307812yes—
407C4C2×#48711yescharacteristic
408C4C2×#48911yescharacteristic
409C4#313911yescharacteristic
410C4#314011yescharacteristic
411C2—24no—
412C2#1878712yes—
413C2#1878912yes—
414C2C2×#202411yescharacteristic
415C2#2345511yescharacteristic
416C2#2346311yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
5. #10355C2semidirect
6. #10349C2semidirect
54. C2×C4⋊Q8Q8semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#349382
#218042
#213132
C2×#19872
#103552
#103492
#80452
#79012
#78912